The usual story is roughly this. The quantifiers of a first-order logic (ordinary universal and existential quantifiers; or perhaps fancier dyadic quantifiers) range over objects in some given domain. A second-order logic, as well as having those first-order quantifiers, has quantifiers ranging over properties of those objects in the given domain -- or what may or may not come to the same thing, ranging over sets of objects from the domain. A third-order logic adds quantifiers ranging over properties of properties of things in the domain -- or what may or may not come to the same thing, ranging over sets of sets of objects from the domain. And so it goes. It's not for nothing, then, that people do often say, following Quine, that higher order logics are just fragments of set theory in disguise. But be that as it may. Where does e.g. propositional modal logic fall into the picture? Well, on the one hand, propositional modal logic has no quantifiers. So a fortiori it doesn't have first...

Is it worth exposing flaws in the reasoning for a position, even if you haven't something better to replace it with? Certainly. At the very least, revealing flaws ought to make proponents less dogmatic in their endorsing of the position: they should proceed with caution in trying to implement policies based on the position, not close off the consideration of counter-proposals, etc. etc. All of which consquences are, in general, surely to be encouraged!

Another hint (and a rather shorter read!!): the opening pages of To mock a mockingbird and other logic puzzles by Raymond M. Smullyan. I couldn't possibly recommend looking via Google Books. :-)

Well, yes of course, two incompetent reasoners could reach different conclusions by making different mistakes! So I take it the question is: could two people reason correctly to different conclusions from the same premisses? But again the answer to that is, trivially, "yes". A given bunch of premisses will entail lots of conclusions (e.g. the axioms of Euclidean geometry entail both that the angles of a triangle add up to two right angles, and that the tangent at the point on a circle is at a right angle to the diameter through that point). One reasoner can correctly deduce one conclusion; another reasoner can correctly deduce a different conclusion. Still, in that case, the different conclusions are all consistent with each other. So perhaps the question is supposed to be: can two two people reason correctly from the same premisses to conclusions that are "different" in the strong sense of being actually incompatible with each other? Again, the answer to that is obviously "yes". For...

A logically valid inference is one that is necessarily truth-preserving. That's pretty much a matter of definition. We just wouldn't count an inference as logically valid if it didn't meet that condition. (In other words, necessarily preserving truth is a necessary condition for an inference to count as valid. Perhaps it is sufficient too; or perhaps rather more is needed for a genuinely valid inference: e.g. some relation of relevance between the premisses and conclusion. But we needn't worry about that. For present purposes it is enough that it is agreed on all sides that to count as valid an inference must at least be truth-preserving.) Now, the "laws of logic" are general principles specifying which types of inference are indeed logically valid. So again, it is pretty much a matter of definition that instances of the laws of logic have to be truth-preserving. That's what it takes for a law to be a law of logic. It isn't as if we can first identify a class of principles as laws of logic...

Suppose I think (a) that it is normally wrong to kill humans, because so doing deprives them of a future life. But I also think (b) women have a "right to choose", and it is permissible to have at least a reasonably early abortion. Then I seem to be in trouble. For by (a) killing a very young human being in utero should be wrong, as it surely deprives it of the long future life it would otherwise have had, while by (b) killing it is permissible. On the face of it, then, my moral views (a) and (b) aren't consistent with each other, but imply that a certain act is both wrong and not wrong -- which is absurd. And note, I can't just shrug my shoulders and cheerfully say "ok, mymoral views are inconsistent" because inconsistent views don't give me anyguidance about what to do, and my moral views are supposed to help guide me! I want to decide to do in various circumstances, and inconsistent moral injunctions are no use at all for deciding. So I need to revise (a) or revise (b), or at least spell out ...

You are quite right that Gödel's (first) incompleteness theorem attracts all kinds of bizarre "interpretations". Various examples are discussed and dissected in Torkel Franzen's very nice short book, Gödel's Theorem: An Incomplete Guide to its Use and Abuse , which I warmly recommend. My guess is that a main source for the whacky interpretations is the claim that has repeatedly been made that the theorem shows that we can't be "machines", and so -- supposedly -- we must be something more than complex biological mechanisms. You can see why that conclusion might in some quarters be found welcome (and other technical results in logic generally don't seem to have such an implication). But as Franzen explains very clearly, it doesn't follow from the theorem.

No. But that's no limitation on such a god's power. We're not saying that there is some possible task that this god fails to be able to pull off. We're saying that there isn't any task that is coherently describable as "making a square triangle". For consider: what could possibly count as making a square triangle? To be a square requires having four sides. To be a triangle requires not having four sides but only three. So nothing can possibly count as being both a square and a triangle. Hence whatever the god (or anyone else) does, it couldn't correctly be described as "making a square triangle" for that isn't a coherent description of anything. Take a mundane case. I pass you the cookies. You can take one. Or you can take none. Both are within your power. But you can't simultaneously both take one and not take one. But saying that plainly isn't to say that there is some limitation on your powers of choice vis-a-vis cookies! The point is that nothing could count as...

I'm reminded of the exasperated Bertrand Russell faced with the young Wittgenstein: "He thinks that nothing empirical is knowable. I asked him to admit that there was not a rhinoceros in the room, but he wouldn't. I looked under all the desks without finding one but Wittgenstein remained unconvinced." It is Wittgenstein here who is being obtuse and in the grip of a silly theory. Of course we can establish empirical propositions both positive and negative – for example, that there are five desks in the room and no rhinoceroses. By any sane standard, it is just plain false that you can't prove a negative, and that supposed "discussion-killer" should itself be promptly killed off.

Logic is about what follows from what. But what follows from what isn't always obvious (or else, e.g., pure maths would be a lot easier than it is). So we need ways of demonstrating unobvious entailments. And a standard way of doing this is to show how we can get from the given premisses to the intended conclusion by a sequence of small steps, each of which is guaranteed to be truth-preserving. If we can break down the big inferential leap from premisses to conclusion into smaller inferential moves, each one of which is evidently valid, that shows the big leap is valid too. Now all this applies equally to informal reasoning -- e.g. derivations or proofs in informal mathematics -- and to formally tidied-up reasoning alike. There's nothing mysterious then about derivations in formal logic. They just do in a regimented way, in some tightly constrained formal language, the sort of thing we usually do in a less regimented way. And given a formal version of a proof, we can read back an informal,...